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Multiple-scale analysis

About: Multiple-scale analysis is a research topic. Over the lifetime, 1360 publications have been published within this topic receiving 27530 citations.


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TL;DR: In this article, the primary and subharmonic simultaneous resonance of a Duffing oscillator with fractional-order derivative was studied, and the effects of nonlinear factor on the system response were analyzed.
Abstract: The primary and subharmonic simultaneous resonance of Duffing oscillator with fractional-order derivative is studied. Firstly, the approximately analytical solution of the resonance is obtained by the method of multiple scales, and the correctness and satisfactory precision of the analytical solution are verified by numerical simulation. Then, the amplitude–frequency curve equation and phase–frequency curve equation are derived from the analytical solution. The stability condition of the steady-state response is obtained by Lyapunov’s first method, and the state switching between two stable periodic orbits is demonstrated. Finally, the effects of nonlinear factor on the system response are analyzed, and the difference between stiffness softening and stiffness hardening system is demonstrated. The influence of fractional-order term on the system is analyzed in depth, and the effect mechanism of fractional-order term is revealed, i.e., the focus and intensity of effect are determined by the order and coefficient of the fractional-order derivative, respectively.

18 citations

Journal ArticleDOI
TL;DR: In this article, the coupled nonlinear differential equations of the non-linear dynamical two-degree-of-freedom vibrating system including quadratic and cubic nonlinearities are studied.

18 citations

Journal ArticleDOI
TL;DR: A section parameter is proposed to describe the microbeam changes in the upper and lower sections and it is shown that the change of section and gap distance can make the vibration soften, harden, and so on.
Abstract: The micro-electro-mechanical system (MEMS) resonator developed based on surface processing technology usually changes the section shape either due to excessive etching or insufficient etching. In this paper, a section parameter is proposed to describe the microbeam changes in the upper and lower sections. The effect of section change on the mechanical properties is studied analytically and verified through numerical and finite element solutions. A doubly-clamped microbeam-based resonator, which is actuated by an electrode on one side, is investigated. The higher-order model is derived without neglecting the effects of neutral plane stretching and electrostatic nonlinearity. Further, the Galerkin method and Newton⁻Cotes method are used to reduce the complexity and order of the derived model. First of all, the influence of microbeam shape and gap variation on the static pull-in are studied. Then, the dynamic analysis of the system is investigated. The method of multiple scales (MMS) is applied to determine the response of the system for small amplitude vibrations. The relationship between the microbeam shape and the frequency response is discussed. Results show that the change of section and gap distance can make the vibration soften, harden, and so on. Furthermore, when the amplitude of vibration is large, the frequency response softening effect is weakened by the MMS. If the nonlinearity shows hardening-type behavior at the beginning, with the increase of the amplitude, the frequency response will shift from hardening to softening behavior. The large amplitude in-well motions are studied to investigate the transitions between hardening and softening behaviors. Finally, the finite element analysis using COMSOL software (COMSOL Inc., Stockholm, Sweden) is carried out to verify the theoretical results, and the two results are very close to each other in the stable region.

18 citations

Journal ArticleDOI
TL;DR: In this paper, the authors compare analytical results pertaining to the escape from a potential well with the classical Duffing's equation with positive linear and negative cubic stiffness terms, i.e., a softening spring characteristic.
Abstract: This brief note compares analytical results pertaining to the escape from a potential well. The system studied is the classical Duffing's equation with positive linear and negative cubic stiffness terms, i.e. a softening spring characteristic. Under gradually increasing periodic forcing, close to resonance, solutions may escape from the potential in which the system is oscillating. Comparison is made between the method of multiple scales and the harmonic balance method, and their stability characteristics investigated using Floquet theory and the Routh-Hurwitz criterion. This type of equation is familiar as an approximation to the motion of a pendulum, and similar types of equations have more recently been used to model the roll response of floating vessels in regular waves.

18 citations

Journal ArticleDOI
TL;DR: In this paper, a two-field Hu-Washitzu variational formulation is used to determine the dispersive elastodynamic behavior of periodic materials. But the method is based on a two field Hu-washitza variational approach and independent displacement and strain fields are employed.

18 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202320
202237
202150
202042
201972
201851